Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In order to test some implementations of numerical solvers for advection-diffusion equations with non-constant coefficients, I'm looking for examples of equations+border and initial conditions of this type which have explicit solutions. Could you propose any or references to such?

share|cite|improve this question

The method of manufactured solution is very simple. If you have your system $F(u)=0$, then you let your solution be e.g. $u=\sin(x^2)+\cos(\exp(y))$, plug into $F$ to get $F(u)=g(x,y)$. Now $F_0=F(u)-g=0$ is your new equation with solution $u$ and corresponding BC.

share|cite|improve this answer
What about boundary conditions? – Artem Oboturov Aug 17 '13 at 8:08
If you have Dirichlet BC then they are equal to the solution $u$ at the boundary, e.g. $u_{x=0}(y)=u(0,y)$. – Gummi F Aug 19 '13 at 18:20

I can suggest you the following studies where you can find the exact solution of the numerical examples.

1) Vit Dolejsi, hp-DGFEM for nonlinear convection-diffusion problems,

2) Melanie Bittl, Dmitri Kuzmin, Roland Becker, The CG1-DG2 method for convection–diffusion equations in 2D,

3) Tie Zhang, Ying Sheng, Superconvergence and gradient recovery for a finite volume element method for solving convection-diffusion equations, DOI: 10.1002/num.21862

I hope they will work...

Best wishes...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.